Mazur-Ulam theorem

Theorem.

Note that we consider isometries to be surjective by definition.
The result is not in general true for non-surjective isometric mappings.

The result does not extend to normed vector spaces over ℂ,
as can be seen from the fact that complex conjugation is an isometry ℂ→ℂ
but is not affine over ℂ.
(But complex conjugation is clearly affine over ℝ,
and in general any normed vector space over ℂ
can be considered as a normed vector space over ℝ,
to which the theorem can be applied.)

This theorem was first proved by Mazur and Ulam.[1]
A simpler proof has been given by Jussi Väisälä.[2]